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Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsimpllr 501 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ps )
 
Theoremsimplrl 502 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ps )
 
Theoremsimplrr 503 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ch )
 
Theoremsimprll 504 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ps )
 
Theoremsimprlr 505 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ch )
 
Theoremsimprrl 506 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ch )
 
Theoremsimprrr 507 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  th )
 
Theoremsimp-4l 508 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  -> 
 ph )
 
Theoremsimp-4r 509 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  ->  ps )
 
Theoremsimp-5l 510 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ph )
 
Theoremsimp-5r 511 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremsimp-6l 512 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ph )
 
Theoremsimp-6r 513 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremsimp-7l 514 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ph )
 
Theoremsimp-7r 515 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremsimp-8l 516 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ph )
 
Theoremsimp-8r 517 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ps )
 
Theoremsimp-9l 518 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  -> 
 ph )
 
Theoremsimp-9r 519 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  ->  ps )
 
Theoremsimp-10l 520 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  -> 
 ph )
 
Theoremsimp-10r 521 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ps )
 
Theoremsimp-11l 522 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  -> 
 ph )
 
Theoremsimp-11r 523 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  ->  ps )
 
Theorempm4.87 524 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
 |-  ( ( ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch )
 ) )  /\  (
 ( ph  ->  ( ps 
 ->  ch ) )  <->  ( ps  ->  (
 ph  ->  ch ) ) ) )  /\  ( ( ps  ->  ( ph  ->  ch ) )  <->  ( ( ps 
 /\  ph )  ->  ch )
 ) )
 
Theorema2and 525 Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
 |-  ( ph  ->  (
 ( ps  /\  rh )  ->  ( ta  ->  th ) ) )   &    |-  ( ph  ->  ( ( ps 
 /\  rh )  ->  ch )
 )   =>    |-  ( ph  ->  (
 ( ( ps  /\  ch )  ->  ta )  ->  ( ( ps  /\  rh )  ->  th )
 ) )
 
Theoremanimpimp2impd 526 Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
 |-  ( ( ps  /\  ph )  ->  ( ch  ->  ( th  ->  et )
 ) )   &    |-  ( ( ps 
 /\  ( ph  /\  th ) )  ->  ( et 
 ->  ta ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  ->  ( ps  ->  ( th  ->  ta ) ) ) )
 
Theoremabai 527 Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ( ph  /\  ps ) 
 <->  ( ph  /\  ( ph  ->  ps ) ) )
 
Theoreman12 528 Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ps  /\  ( ph  /\  ch ) ) )
 
Theoreman32 529 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ps ) )
 
Theoreman13 530 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ch  /\  ( ps  /\  ph ) ) )
 
Theoreman31 531 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ch  /\  ps )  /\  ph )
 )
 
Theoreman12s 532 Swap two conjuncts in antecedent. The label suffix "s" means that an12 528 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ps  /\  ( ph  /\  ch )
 )  ->  th )
 
Theoremancom2s 533 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ch  /\  ps ) ) 
 ->  th )
 
Theoreman13s 534 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ch  /\  ( ps  /\  ph )
 )  ->  th )
 
Theoreman32s 535 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremancom1s 536 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph )  /\  ch )  ->  th )
 
Theoreman31s 537 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ch 
 /\  ps )  /\  ph )  ->  th )
 
Theoremanass1rs 538 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremanabs1 539 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  <->  (
 ph  /\  ps )
 )
 
Theoremanabs5 540 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
 |-  ( ( ph  /\  ( ph  /\  ps ) )  <-> 
 ( ph  /\  ps )
 )
 
Theoremanabs7 541 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 ) 
 <->  ( ph  /\  ps ) )
 
Theoremanabsan 542 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
 |-  ( ( ( ph  /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss1 543 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss4 544 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
 |-  ( ( ( ps 
 /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss5 545 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ph  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabsi5 546 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ph  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi6 547 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
 |-  ( ph  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi7 548 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ps  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi8 549 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
 |-  ( ps  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabss7 550 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremanabsan2 551 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ps  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss3 552 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoreman4 553 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( ps  /\  th ) ) )
 
Theoreman42 554 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( th  /\  ps ) ) )
 
Theoreman4s 555 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  th ) ) 
 ->  ta )
 
Theoreman42s 556 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( th  /\  ps ) ) 
 ->  ta )
 
Theoremanandi 557 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandir 558 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ( ps  /\  ch ) ) )
 
Theoremanandis 559 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ta )
 
Theoremanandirs 560 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ta )
 
Theoremsyl2an2 561 syl2an 283 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ch  /\  ph )  ->  th )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  (
 ( ch  /\  ph )  ->  ta )
 
Theoremsyl2an2r 562 syl2anr 284 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ph  /\  ch )  ->  th )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ch )  ->  ta )
 
Theoremimpbida 563 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  ps )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm3.45 564 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  ch )
 ) )
 
Theoremim2anan9 565 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( ph  /\  th )  ->  ( ( ps 
 /\  ta )  ->  ( ch  /\  et ) ) )
 
Theoremim2anan9r 566 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  ->  ( ch  /\  et )
 ) )
 
Theoremanim12dan 567 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ( ch  /\  ta )
 )
 
Theorempm5.1 568 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
 |-  ( ( ph  /\  ps )  ->  ( ph  <->  ps ) )
 
Theorempm3.43 569 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\ 
 ch ) ) )
 
Theoremjcab 570 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
 |-  ( ( ph  ->  ( ps  /\  ch )
 ) 
 <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
 
Theorempm4.76 571 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  /\  ch )
 ) )
 
Theorempm4.38 572 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theorembi2anan9 573 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  /\  ta ) 
 <->  ( ch  /\  et ) ) )
 
Theorembi2anan9r 574 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et )
 ) )
 
Theorembi2bian9 575 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  <->  ta )  <->  ( ch  <->  et ) ) )
 
Theorempm5.33 576 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ps  ->  ch ) )  <->  ( ph  /\  (
 ( ph  /\  ps )  ->  ch ) ) )
 
Theorempm5.36 577 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ph 
 <->  ps ) )  <->  ( ps  /\  ( ph  <->  ps ) ) )
 
Theorembianabs 578 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
1.2.5  Logical negation (intuitionistic)
 
Axiomax-in1 579 'Not' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Axiomax-in2 580 'Not' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01 581 Reductio ad absurdum. Theorem *2.01 of [WhiteheadRussell] p. 100. This is valid intuitionistically (in the terminology of Section 1.2 of [Bauer] p. 482 it is a proof of negation not a proof by contradiction); compare with pm2.18dc 788 which only holds for some propositions. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( ( ph  ->  -.  ph )  ->  -.  ph )
 
Theorempm2.21 582 From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (Contributed by Mario Carneiro, 12-May-2015.)
 |-  ( -.  ph  ->  (
 ph  ->  ps ) )
 
Theorempm2.01d 583 Deduction based on reductio ad absurdum. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  -.  ps )
 )   =>    |-  ( ph  ->  -.  ps )
 
Theorempm2.21d 584 A contradiction implies anything. Deduction from pm2.21 582. (Contributed by NM, 10-Feb-1996.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theorempm2.21dd 585 A contradiction implies anything. Deduction from pm2.21 582. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ch )
 
Theorempm2.24 586 Theorem *2.24 of [WhiteheadRussell] p. 104. (Contributed by NM, 3-Jan-2005.)
 |-  ( ph  ->  ( -.  ph  ->  ps )
 )
 
Theorempm2.24d 587 Deduction version of pm2.24 586. (Contributed by NM, 30-Jan-2006.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  ( -.  ps  ->  ch ) )
 
Theorempm2.24i 588 Inference version of pm2.24 586. (Contributed by NM, 20-Aug-2001.) (Revised by Mario Carneiro, 31-Jan-2015.)
 |-  ph   =>    |-  ( -.  ph  ->  ps )
 
Theoremcon2d 589 A contraposition deduction. (Contributed by NM, 19-Aug-1993.) (Revised by NM, 12-Feb-2013.)
 |-  ( ph  ->  ( ps  ->  -.  ch )
 )   =>    |-  ( ph  ->  ( ch  ->  -.  ps )
 )
 
Theoremmt2d 590 Modus tollens deduction. (Contributed by NM, 4-Jul-1994.)
 |-  ( ph  ->  ch )   &    |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremnsyl3 591 A negated syllogism inference. (Contributed by NM, 1-Dec-1995.) (Revised by NM, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ch  ->  -.  ph )
 
Theoremcon2i 592 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 28-Nov-2008.) (Proof shortened by Wolf Lammen, 13-Jun-2013.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ps  ->  -.  ph )
 
Theoremnsyl 593 A negated syllogism inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  ps )   &    |-  ( ch  ->  ps )   =>    |-  ( ph  ->  -.  ch )
 
Theoremnotnot 594 Double negation introduction. Theorem *2.12 of [WhiteheadRussell] p. 101. This one holds for all propositions, but its converse only holds for decidable propositions (see notnotrdc 789). (Contributed by NM, 28-Dec-1992.) (Proof shortened by Wolf Lammen, 2-Mar-2013.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremnotnotd 595 Deduction associated with notnot 594 and notnoti 609. (Contributed by Jarvin Udandy, 2-Sep-2016.) Avoid biconditional. (Revised by Wolf Lammen, 27-Mar-2021.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  -.  -.  ps )
 
Theoremcon3d 596 A contraposition deduction. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 31-Jan-2015.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( -.  ch  ->  -.  ps ) )
 
Theoremcon3i 597 A contraposition inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 20-Jun-2013.)
 |-  ( ph  ->  ps )   =>    |-  ( -.  ps  ->  -.  ph )
 
Theoremcon3rr3 598 Rotate through consequent right. (Contributed by Wolf Lammen, 3-Nov-2013.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( -.  ch  ->  ( ph  ->  -.  ps ) )
 
Theoremcon3dimp 599 Variant of con3d 596 with importation. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ( ph  /\ 
 -.  ch )  ->  -.  ps )
 
Theorempm2.01da 600 Deduction based on reductio ad absurdum. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ph  /\  ps )  ->  -.  ps )   =>    |-  ( ph  ->  -.  ps )
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